About an ether interpretation for the Einstein equations of general relativity I. Schmelzer Following Bell's paper \how to teach special relativity", we argue that a generalization of the Lorentz ether interpretation of the Einstein equations of general relativity may have a similar ped- agogical value as the Lorentz ether interpretation for special relativity: It will give better intuitions to those who learn it. The Lorentz ether interpretation used here is the limit of an alternative ether theory of gravity, which gives the Einstein equations of GR in harmonic coordinates, together with an interpretation of the gravitational field in terms of density, velocity and stress tensor of an ether. Different examples where this \Lorentz ether" gives better, less confusing intuitions than a curved spacetime are considered. It is argued that using the Lorentz ether would decrease the problems with \ether cranks", giving them the possibility to accept GR physics without giving up their prejudices about \true" space and time. The limitations of the Lorentz ether { which requires a fixed background { can be used to obtain a better understanding of the revolutionary character of the spacetime interpretation and background independence of GR. Keywords: gravity, Lorentz ether I. PREFACE guish physics (what all interpretations share) from meta- physics (what can differ). Then, different interpretations Many people think that the Michelson-Morley experi- have very different problems, and attempts to solve them ment was the death penalty not only for particular ether provide different starting points for further theory devel- theories, but the ether in general. Much less people know opment. This point is illustrated by an ether theory of that there is the Lorentz ether, an ether interpretation of gravity proposed by the author [14] with the aim to im- special relativity (SR), which predicts exactly the same as prove this ether interpretation. SR for the Michelson-Morley experiment as well as every Other points are specific to the ether interpretation. other experiment. A few people have heard about Ein- We will find that the ether interpretation suggests in stein's Leyden lecture [7], where Einstein embraced the many questions intuitions quite different from those sug- notion of the ether even for general relativity (GR). But it gested by the spacetime interpretion, and those different is almost unknown that the Lorentz ether interpretation intuitions often appear more adequate. This supports of special relativity can be generalized in a quite simple Bell's \impression that those with a more classical edu- way into a Lorentz ether interpretation of the Einstein cation, knowing something of the reasoning of Larmor, equations of general relativity in harmonic coordinates. Lorentz, and Poincare, as well as that of Einstein, have In this interpretation, the gravitational field describes the stronger and sounder instincts" [3]. density, the velocity and the stress tensor of some ether, A specific advantage of the ether interpretation is that and the harmonic coordinate conditions are interpreted it is, contrary to the curved spacetime, not in conflict as the continuity and Euler equations for this ether. One with common sense and classical Newtonian concepts aim of this paper is simply to give an introduction to this about space and time. This allows to teach the math- Lorentz ether interpretation of relativistic gravity. ematics and physics of relativistic gravity without a ne- This includes a consideration of the main difference cessity to introduce any new ideas about space and time. between the Lorentz ether and the spacetime interpreta- Even the key point of the spacetime interpretation { the tion { the existence of a background. This background unobservability of the background { can be teached with- has several consequences: It excludes solutions with non- out questioning its very existence, but, as done in the trivial topopology, as well as with causal loops, and leads ether interpretation, by explaining the unobservability to different notions of completeness and of homogeneity. by distortions of the rulers and clocks by the ether. This At least in principle, this makes the ether interpretation is not in conflict with common sense, which is used to even a different physical theory: Observing a nontrivial handle distortions of measurement instruments. Then it topology or a causal loop would falsify the ether inter- can be left to decision of the pupils if they prefer to fol- pretation, but not the spacetime interpretation. low the common sense compatible ether interpretation, The focus of this paper is, instead, a pedagogical or accept that, once the background does not have any one, following Bell's remark that \we need not accept directly observable consequences, it makes no sense to Lorentz's philosophy to accept a Lorentzian pedagogy" assume its existence too. [3]. The main point is that, independent of the arguments This would allow to solve the problems of \ether in favor or against a particular interpretation, it would cranks", who reject even special relativity as \logically be useful to know (and to teach) both interpretations. wrong" and explain the success of relativistic physics Some of the arguments have general character: Know- with anti-scientific conspiracy theories. What they reject ing different interpretations allows much better to distin- is, essentially, the spacetime interpretation. Rejecting 2 the spacetime interpretation means, at least this is how III. THE PEDAGOGIC VALUE OF THE it looks today, to reject relativistic physics completely. MATHEMATICS OF THE ETHER If they would know that relativistic physics is compati- INTERPRETATION ble with classical ether metaphysics, they probably would not become anti-scientific cranks, but, instead, follow the The mathematics required in the ether interpretation ether interpretation and reject only the spacetime inter- are important, useful to be teached anyway. The har- pretation. monic coordinates, proposed by de Donder [5] and Lanc- zos [10], simplify the Einstein equations in a very essen- tial way: They obtain the form mn ab mn mn pq pq II. THE LORENTZ ETHER INTERPRETATION G = g @a@bg + F (g ;@kg ) (5) OF THE EINSTEIN EQUATIONS without any mixing of different components of gmn in the second order derivative terms. They are probably the The equations of the Lorentz ether interpretation are most often used coordinate condition in GR. In particu- the Einstein equations of GR in harmonic coordinates. lar, they play a key role in \unreasonable effective" post- The harmonic coordinate condition is, different from the Newtonian approximation [19]. Based on this important spacetime interpretation of GR, not only a quite arbi- simplification, Fock [8] has argued in favor of harmonic trary coordinate condition without any physical impor- coordinates being natural preferred coordinates. Indeed, tance, but an additional physical equation. It distin- what makes classical Cartesian coordinates preferred in 0 i guishs the preferred coordinates x = t; x of a Newtonian classical physics is that the equations of physics obtain background: an especially simple form. The same can be said for har- monic coordinates too. One important argument in this direction is that the ν µν p x = @µ(g −g) = 0: (1) first local existence and uniqueness theorems for the Ein- stein equations have been proven by Bruhat in harmonic coordinates [4]. The point mentioned above that the Given that the harmonic condition has already the highest order terms no longer mix in harmonic coordi- form of four conservation laws for a symmetric tensor, it nates played the decisive role here. seems natural to compare it with the conservation laws The other part of the mathematical apparatus used in we know from condensed matter theory { the continuity the Lorentz ether interpretationp is the decomposition of and Euler equations: the gravitational field gmn −g into a scalar field (ρ), a three-vector field (vi), and a three-metric (σij), which is i @tρ + @iρv = 0 (2) the result of the choice of one time-like harmonic coordi- j i j ij nate as the time. This is a variant of the ADM decom- @tρv + @i(ρv v + σ ) = 0 (3) position, which was necessary to define a Hamiltonian i ij formulation for GR [1]. for density ρ, velocity v and stress tensor σ of some So, even if teaching the Lorentz ether interpretation condensed matter. This suggests the following identifica- would require some additional mathematics, all these ad- tions: ditional elements { the harmonic coordinate condition as well as the ADM decomposition { are important con- 00p 0i 00 i ijp i j ij g −g = ρ, g =g = v ; g −g = ρv v + σ : tributions to the mathematics of GR, are proposed and (4) widely used without any connection to the Lorentz ether, To obtain a positive density ρ > 0, all we have to require and have their own value. This allows to make two is that x0 = t is time-like. If the other coordinates xi points: On the one hand, teaching the Lorentz ether are space-like, the stress tensor σij would be negative- gives a nice motivation to teach these additional mathe- definite. matics. On the other hand, all the mathematics used in As in the special-relativist Lorentz ether, the preferred the Lorentz ether have been developed and widely used coordinates define a classical Newtonian world with ab- independently, which suggests that the interpretation is solute space and time. The spatial coordinates xi are in itself quite natural. standard Cartesian coordinates of absolute space, and the time coordinate t defines absolute time. Similarly we are unable to measure absolute distances and times with IV. THE LIMITATIONS OF THE LORENTZ our rulers and clocks, because they are distorted by the ETHER ether: What they measure are the distances defined by the spacetime metric gµν (x; t). An important difference It is not without reason that I have named the Lorentz is that the ether is no longer motionless, homogeneous, ether an \interpretation of the Einstein equations" in- and incompressible, but follows the usual equations for stead of an \interpretation of GR".